Consider a system with stochastic reaction events

$$x \stackrel{\lambda}{\longrightarrow} x+1 \quad \text { and } \quad x \stackrel{\beta x^{2}}{\longrightarrow} x-2,$$

where $\lambda$ and $\beta$ are rate constants.

(a) State or derive the exact differential equation satisfied by the average number of molecules $<x>$. Assuming that fluctuations are negligible, approximate the differential equation to obtain the steady-state value of $\langle x\rangle$.

(b) Using this approximation, calculate the elasticity $H$, the average lifetime $\tau$, and the average chemical event size $<r>$ (averaged over fluxes).

(c) State the stationary Fluctuation Dissipation Theorem for the normalised variance $\eta$. Hence show that

$$\eta=\frac{3}{4<x>} .$$